Section 1.8

• 20. a) O(n^3 log n); b) O(2^2 3^n); c) O(n^n n!)
• 22.
  • a) x <= 3x+7 <= 4x, for x > 7;
  • b) x^2 <= 2x^2+x-7 <= 3x^2, x>7;
  • c) 1/2 x <= floor(x+1/2) <= 2x, for x > 2;
  • d) log2 x <= log(x^2+1) <= log (x^2+x^2) = log(2x^2) = log 2 + log x^2 = 1 + 2log x <= 3 log x;
  • e) 1/log2 10 log2 x <= log10 x <= 1/log2 10 log2 x
• 24. 3x^2 <= 3x^2 + x + 1 <= 3x^2 + x^2 = 4x^2, x>2
• 26. 1(3)(5)...(2n-1) <= 2n(2n)(2n)(2n)...(2n) = (2n)^n. So O((2n)^n).
• 28. f(x) <= C logb x = C (1/loga b) loga x = D loga x.
• 30. No. Try f(x) = 2x and g(x) = x.
• 32. a) lim cf(x)/g(x) = c lim f(x)/g(x) = c(0) = 0. b) lim ((f1(x)+f2(x))/g(x)) = lim (f1(x)/g(x) + f2(x)/g(x)) = lim f1(x)/g(x) + lim(f2(2)/g(x)) = 0+0 = 0.
• 36. If f(x) = x and g(x) = x^2 , f(x) is o(g(x)), but log|f(x)| is not o(log|g(x)|).
• 38. lim (poly of degree n / poly of degree m) where m > n. Divide numerator and denominator by x^n. Numerator is bounded, denominator is not, so limit as x->inf is 0.
• 40. 1 + Sum(j=2,n) 1/j < 1 + Integral(1,n) 1/x dx = 1 + ln n = 1 + C log n <= 2C log n.
• 42. Sum(i=1,n) log(i) < Sum(i=1,n) log(n) = n log n